Sure. If f1(x), f2(x) and f3(x) are linearly dependent, then there are nonzero constants such that c1*f1(x)+c2*f2(x)+c3*f3(x) is identically zero over some interval. So a linear combination of columns in your matrix is zero. This tells you det=0 over the interval. So if det is non-zero anywhere, you know they aren't linearly dependent.